I have tracked down some results on explicit classifications of simple modules for $u_q(\mathfrak{sl}_3)$.  The general picture is that the simple modules are bigraded by the root lattice and look like towers of concentric hexagons.

For the benefit of anyone else interested, there is a long series of papers by [Dobrev][1]:

Multiplet classification of highest weight modules over quantum universal enveloping algebras: the Uq(SL(3,C)) example in Groups St Andrews 1989 vol 1 LMS LNM #159

Representations of Quantum Groups, Symmetries in Science V (Lochau 1990), 93–135, Plenum Press, NY, 1991.

A chapter from Lecture Notes in Physics, 1990, Volume 370, [here][2]

Dobrev-Truini [Irregular Uq(sl(3)) representations at roots of unity via Gel’fand–(Weyl)–Zetlin basis][3]

Dobrev-Truini [Polynomial realization of the Uq(sl(3)) Gel’fand–(Weyl)–Zetlin basis][4]

[MR1182163][5], [MR1191199][6], 

...and many others.

also there is a paper by
[Abdesselam, Arnaudon, Chakrabarti][7]
and a discussion of dimensions by Pereira [here][8]

Some of the relevant material is hard to find and/or requires paying large sums of money to publishing corporations.


  [1]: http://theo.inrne.bas.bg/~dobrev/
  [2]: http://www.springerlink.com/content/7h0t2860r175956r/
  [3]: http://jmp.aip.org/resource/1/jmapaq/v38/i5/p2631_s1
  [4]: http://jmp.aip.org/resource/1/jmapaq/v38/i7/p3750_s1
  [5]: http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=58600&vfpref=html&r=118&mx-pid=1182163
  [6]: http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=58600&vfpref=html&r=111&mx-pid=1191199
  [7]: http://www.citebase.org/abstract?id=q-alg%2F9504006
  [8]: http://www.cmat.edu.uy/~mariana/dobrev.pdf